Let $p,q$ be positive reals with sum 1. Show that for any $n$-tuple of reals $(y_1,y_2,...,y_n)$, there exists an $n$-tuple of reals $(x_1,x_2,...,x_n)$ satisfying $$p\cdot \max\{x_i,x_{i+1}\} + q\cdot \min\{x_i,x_{i+1}\} = y_i$$for all $i=1,2,...,2017$, where $x_{2018}=x_1$.
Problem
Source: 2018 China TST Day 1 Q1
Tags: algebra
omarius
03.01.2018 00:15
sccdgsy wrote:
Intermediate value theorem
To #1: In the original problem, n=2017. How ?
l1090107005
05.01.2018 08:18
Is that $n$ equals to 2017?
liekkas
06.01.2018 09:39
The statement is true for any odd $n$. Just to notice that once $x_1$ is fixed, we can find a unique $x_2$ such that $p\cdot \max\{x_1,x_2\} + q\cdot \min\{x_1,x_2\} = y_1$. By applying same operation we can get a sequence $x_1,x_2,...,x_{2017},f(x_1)$, where $f(x_1)$ is generated by $x_{2017}$. Now notice that when $x_1$ tends to $+\infty$ ,$x_2$ tends to $-\infty$, then $f(x_1)$ tends to $-\infty$, and vice versa. Then we can easily find a $x_1 = f(x_1)$ for $f$ is continuous and decreasing. Remarks: for any even $n$ the statement is false. Indeed, $y_k=a$ for odd $k$ and $y_k=b$ for even $k$,where $a \neq b$ is a counterexample.
Zawadx
13.01.2018 19:31
liekkas wrote: The statement is true for any odd $n$. Just to notice that once $x_1$ is fixed, we can find a unique $x_2$ such that $p\cdot \max\{x_1,x_2\} + q\cdot \min\{x_1,x_2\} = y_1$. By applying same operation we can get a sequence $x_1,x_2,...,x_{2017},f(x_1)$, where $f(x_1)$ is generated by $x_{2017}$. Now notice that when $x_1$ tends to $+\infty$ ,$x_2$ tends to $-\infty$, then $f(x_1)$ tends to $-\infty$, and vice versa. Then we can easily find a $x_1 = f(x_1)$ for $f$ is continuous and decreasing. Remarks: for any even $n$ the statement is false. Indeed, $y_k=a$ for odd $k$ and $y_k=b$ for even $k$,where $a \neq b$ is a counterexample. In addition to this, note that when $x_i=y_i$ we have $x_{i+1}=y_i$ too. Since $x_{i+1}$ varies monotonically on either side of $x_i=y_i$, we have $x_{i+1}$ covering all of $\mathbb{R}$. So the above holds.
HolyMath
05.02.2018 12:08
very good !!!
ywq233
28.02.2020 13:23
The original problem also asked to show the uniqueness of tuple $(x_1,x_2,...,x_{2017})$.
menpo
03.07.2020 09:38
Can someone explain in details?
guptaamitu1
03.05.2022 02:35
ywq233 wrote: The original problem also asked to show the uniqueness of tuple $(x_1,x_2,...,x_{2017})$. BUMP. How to show the uniqueness of tuple?